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Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. (English) Zbl 0808.35128
The well-posedness, which includes existence, uniqueness, and continuous dependence upon initial data, is studied for the initial-value problem for the generalized Korteweg-de Vries (KdV) equations of the form \[ u_ t+ u_{xxx}+ u^ k u_ x=0, \tag{1} \] where \(k\) is a positive integer. It is well known that equation (1) is exactly integrable for \(k=1\) (the classical KdV equation) and for \(k=2\) (the modified KdV equation). At any value of \(k\), equation (1) has two fundamental integrals of motion, the “momentum” \(\int_{-\infty}^{+\infty} u^ 2(x)dx,\) and the energy (Hamiltonian). These two integrals of motion are essentially used in the well-posedness analysis. The analysis is based on global estimates for an explicit solution of the linear initial-value problem associate to equation (1), \(v_ t+ v_{xxx}=0\) combined with the contraction mapping principle. Introducing special functional norms, the authors demonstrate that, for each particular value of \(k\), there is a relevant class of the Sobolev space to which the initial data \(u_ 0(x)\equiv u(x,t=0)\) must belong in order to allow for the well-posedness proof, the existence being proved for finite times. The technique of the proof and the particular results obtained are essentially different for the cases \(k<4\), \(k=4\), and \(k>4\), which reflects the fundamental property of equation (1): while at \(k<4\) the solution developing from generic initial data remains smooth indefinitely long, the weak and strong collapse sets in at a finite time, respectively, at \(k=4\) and at \(k>4\).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35P25 Scattering theory for PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B45 A priori estimates in context of PDEs
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